Hodge Dual - Explanation

Explanation

Let W be a vector space, with an inner product . For every linear function there exists a unique vector v in W such that for all w in W. The map given by is an isomorphism. This holds for all vector spaces, and can be used to explain the Hodge dual.

Let V be an n-dimensional vector space with basis . For 0 ≤ k≤ n, consider the exterior power spaces and . For each and, we have . There is, up to a scalar, only one n-vector, namely . In other words, must be a scalar multiple of for all and .

Consider a fixed . There exists a unique linear function such that for all . This is the scalar multiple mentioned in the previous paragraph. If denotes the inner product on (n–k)-vectors, then there exists a unique (n–k)-vector, say, such that for all . This (n–k)-vector is the Hodge dual of λ, and is the image of the under the canonical isomorphism between and . Thus, .

Read more about this topic:  Hodge Dual

Famous quotes containing the word explanation:

    “My companion assumes to know my mood and habit of thought, and we go on from explanation to explanation, until all is said that words can, and we leave matters just as they were at first, because of that vicious assumption.”
    —Ralph Waldo Emerson (1803–1882)

    “The explanation of the propensity of the English people to portrait painting is to be found in their relish for a Fact. Let a man do the grandest things, fight the greatest battles, or be distinguished by the most brilliant personal heroism, yet the English people would prefer his portrait to a painting of the great deed. The likeness they can judge of; his existence is a Fact. But the truth of the picture of his deeds they cannot judge of, for they have no imagination.”
    —Benjamin Haydon (1786–1846)

    “We live between two worlds; we soar in the atmosphere; we creep upon the soil; we have the aspirations of creators and the propensities of quadrupeds. There can be but one explanation of this fact. We are passing from the animal into a higher form, and the drama of this planet is in its second act.”
    —W. Winwood Reade (1838–1875)